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\begin{document}
\begin{center}
{\bf \large Math 701 Problem Set 11} \vs \\
{\bf due Friday 12/13/2013} \vs \vs
\end{center}

\noindent
$\fM$ and $m$ will denote the $\sigma$-algebra of Lebesgue measurable sets in $\RR$ and Lebesgue measure, respectively. \vs \vs   

\noindent
{\bf Problem 1.} 
\begin{enumerate}
\item[(i)]
Suppose $f,g : \RR \to [-\infty,+\infty]$ are measurable. Show that the sets
$$
\{ x \in \RR: f(x)<g(x) \} \qquad \text{and} \qquad \{ x \in \RR: f(x)=g(x) \}
$$
are measurable. \vs
\item[(ii)]
Suppose $f_n :\RR \to [-\infty,+\infty]$ is a sequence of measurable functions. Show that the set of points at which $\lim_{n \to \infty} f_n$ exists is measurable. \vs
\end{enumerate} 

\noindent
{\bf Problem 2.} Let $C \subset [0,1]$ be the middle-thirds Cantor set and $f:C \to [0,1]$ be the standard surjection which sends ternary to binary expansions. $f$ extends to a continuous non-decreasing map $f:[0,1] \to [0,1]$ which is constant on every gap of $C$ (this is the so-called ``Cantor function'' or ``Devil's staircase''). Let $g(x)=x+f(x)$. Show that $g$ is continuous and strictly increasing, and $g(C) \subset [0,2]$ has measure $1$. Use this to prove that there are measurable subsets of $C$ which map continuously to non-measurable sets (all such measurable sets are therefore non-Borel!). \vs

\noindent
{\bf Problem 3.} Suppose $f \in L^1(\RR)$. Show that there is a constant $C>0$ such that for all $t>0$, 
$$
m \big( \{ x \in \RR : |f(x)| > t \} \big) < \frac{C}{t}. 
$$   

\noindent
{\bf Problem 4.} The {\bit average value} of $f \in L^1(\RR)$ over a set $E \in \fM$ of positive measure is defined by
$$
\bar{f}_E = \frac{1}{m(E)} \int_E f \, dm.
$$
Prove that $\bar{f}_E \in [a,b]$ for every $E$ if and only if $f(x)\in [a,b]$ for almost every $x \in \RR$. \vs

\noindent
{\bf Problem 5.} Suppose $f_n: \RR \to [0,+\infty]$ are measurable, $f_1 \geq f_2 \geq f_3 \geq \cdots \geq 0$, and $f_n(x) \to f(x)$ for all $x \in \RR$. If $f_1 \in L^1(\RR)$, show that 
$$
\lim_{n \to \infty} \int_\RR f_n \, dm = \int_\RR f \, dm. 
$$
Show by an example that the condition $f_1 \in L^1(\RR)$ cannot be dispensed with. \vs 

\noindent
{\bf Problem 6.} Suppose $f \in L^1(\RR)$. Show that for every $\ve>0$ there is a $\delta>0$ such that $m(E)<\delta$ implies $\int_E |f| \, dm <\ve$. (Hint: Proceed in one of the following ways: (1) Work with the sequence $f_n=\min \{ |f|,n \}$ of bounded integrable functions on $\RR$ which converges monotonically to $|f|$; or (2) Suppose there is an $\ve>0$ and sets $E_n$ with $m(E_n)<1/2^n$ such that $\int_{E_n} |f| \, dm > \ve$ for all $n$. Look at the integral of $|f|$ over $\bigcup_{k \geq n} E_k$ as $n \to \infty$ to reach a contradiction.)      




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